Propagating Error a Pendulum

[nubey1]

Homework Statement

The length of a string attached to a pendulum is measured with a precision of (+or-)0.2. The time of the oscillation is measured to a precision of (+or-)0.1. How many periods must you measure so that the contribution of the uncertainty in time is smaller than the uncertainty in length, when calculating g?

Homework Equations

T=2pi(l/g)^(1/2)

The Attempt at a Solution

g=((2pi)^2(l+delta:l))/(T+delta:T)^2
I don't know where to go from here.
Delta l and T are the error in those measurements.

 

[dark adonis]

Homework Statement

The length of a string attached to a pendulum is measured with a precision of (+or-)0.2. The time of the oscillation is measured to a precision of (+or-)0.1. How many periods must you measure so that the contribution of the uncertainty in time is smaller than the uncertainty in length, when calculating g?

Homework Equations

T=2pi(l/g)^(1/2)

The Attempt at a Solution

g=((2pi)^2(l+delta:l))/(T+delta:T)^2
I don't know where to go from here.
Delta l and T are the error in those measurements.

The next thing to do is assume that the error is really small relative to the actual value.
$$\delta l << l$$ and $$\delta T << T$$. I think you might have an error in your equation for g:
$$g +\delta g= (2 \pi)^2 \frac{l+\delta l }{(T+\delta T)^2}$$ where $$\delta g$$ is the error in g.
If you are familiar with calculus then this comes out to:
$$\delta g = |\frac{\partial g}{\partial l}| \delta l + |\frac{\partial g}{\partial T}| \delta T$$
If you don't have the luxury of calculus we might need to know what relations for uncertainty you are given to clue you in

 

[nlightner]

I understand the importance of accurately measuring and minimizing error in experiments. In this case, we are looking at the measurement of a pendulum's length and time of oscillation and how it affects the calculation of g, the acceleration due to gravity. The length of the string is measured with a precision of (+/-) 0.2 and the time of oscillation is measured with a precision of (+/-) 0.1.

To determine the number of periods needed to minimize the contribution of uncertainty in time, we can use the equation T=2pi(l/g)^(1/2). This equation shows that g is directly proportional to the square of the time (T) and inversely proportional to the square root of the length (l). Therefore, to minimize the contribution of uncertainty in time, we want to increase the number of periods measured, which will result in a larger T value.

To determine the minimum number of periods needed, we can use the equation for error propagation: delta g = ((2pi)^2(l+delta:l))/(T+delta:T)^2. Here, delta l and delta T represent the error in the measurements of length and time, respectively.

To make the contribution of uncertainty in time smaller than the uncertainty in length, we want delta g to be smaller than delta l. This means we need to increase the value of T until delta g is smaller than delta l. This can be achieved by increasing the number of periods measured until the value of delta g is smaller than delta l.

In conclusion, to minimize the contribution of uncertainty in time when calculating g, we need to measure enough periods so that the value of delta g is smaller than delta l. This can be achieved by increasing the number of periods measured until the value of delta g is smaller than delta l.